Question: Solve for $x$ and $y$ using elimination. $\begin{align*}4x-4y &= -6 \\ -5x+2y &= -2\end{align*}$
Solution: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $1$ and the bottom equation by $2$ $\begin{align*}4x-4y &= -6\\ -10x+4y &= -4\end{align*}$ Add the top and bottom equations. $-6x = -10$ Divide both sides by $-6$ and reduce as necessary. $x = \dfrac{5}{3}$ Substitute $\dfrac{5}{3}$ for $x$ in the top equation. $4( \dfrac{5}{3})-4y = -6$ $\dfrac{20}{3}-4y = -6$ $-4y = -\dfrac{38}{3}$ $y = \dfrac{19}{6}$ The solution is $\enspace x = \dfrac{5}{3}, \enspace y = \dfrac{19}{6}$.